On-Line Searching in Simple Polygons

In this paper we study the problem of a robot searching for a visually recognizable target in an unknown simple polygon. We present two algorithms. Both work for arbitrarily oriented polygons. The search cost is proportional to the distance traveled by the robot. We use competitive analysis to judge the performance of our strategies. The rst one is a simple modiication of Djikstra's shortest path algorithm and achieves a competitive ratio of 2n ? 7 where n is the number of vertices of the polygon. The second strategy achieves a competitive ratio of 1 + 2 (2k) 2k =(2k ? 1) 2k?1 if the polygon is k-spiral. This can be shown to be optimal. It is based on an optimal algorithm to search in geometric trees which is also presented in this paper. Consider a geometric tree with m leaves and a target hidden somewhere in the tree. The target can only be detected if the robot reaches the location of the target. We provide an algorithm that achieves a competitive ratio of 1 + 2m m =(m ? 1) m?1 and show its optimality.